I solved this same problem; my wife and I play the game together, but I wanted to do a solo mission without messing up our achievements.

I did it by using Steam Family Sharing. They allow you to register a second Steam profile on the same computer, and then share games between them. Logging into my new account, I requested access to the game (which I granted to myself) and it started up with a fresh set of achievements.

When I want to go back to the other set of achievements, I use the other profile.